3.3.93 \(\int \frac {(a+b x^3)^8}{x^4} \, dx\) [293]

Optimal. Leaf size=105 \[ -\frac {a^8}{3 x^3}+\frac {28}{3} a^6 b^2 x^3+\frac {28}{3} a^5 b^3 x^6+\frac {70}{9} a^4 b^4 x^9+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{15} a^2 b^6 x^{15}+\frac {4}{9} a b^7 x^{18}+\frac {b^8 x^{21}}{21}+8 a^7 b \log (x) \]

[Out]

-1/3*a^8/x^3+28/3*a^6*b^2*x^3+28/3*a^5*b^3*x^6+70/9*a^4*b^4*x^9+14/3*a^3*b^5*x^12+28/15*a^2*b^6*x^15+4/9*a*b^7
*x^18+1/21*b^8*x^21+8*a^7*b*ln(x)

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Rubi [A]
time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^8}{3 x^3}+8 a^7 b \log (x)+\frac {28}{3} a^6 b^2 x^3+\frac {28}{3} a^5 b^3 x^6+\frac {70}{9} a^4 b^4 x^9+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{15} a^2 b^6 x^{15}+\frac {4}{9} a b^7 x^{18}+\frac {b^8 x^{21}}{21} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^4,x]

[Out]

-1/3*a^8/x^3 + (28*a^6*b^2*x^3)/3 + (28*a^5*b^3*x^6)/3 + (70*a^4*b^4*x^9)/9 + (14*a^3*b^5*x^12)/3 + (28*a^2*b^
6*x^15)/15 + (4*a*b^7*x^18)/9 + (b^8*x^21)/21 + 8*a^7*b*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (28 a^6 b^2+\frac {a^8}{x^2}+\frac {8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{3 x^3}+\frac {28}{3} a^6 b^2 x^3+\frac {28}{3} a^5 b^3 x^6+\frac {70}{9} a^4 b^4 x^9+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{15} a^2 b^6 x^{15}+\frac {4}{9} a b^7 x^{18}+\frac {b^8 x^{21}}{21}+8 a^7 b \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 105, normalized size = 1.00 \begin {gather*} -\frac {a^8}{3 x^3}+\frac {28}{3} a^6 b^2 x^3+\frac {28}{3} a^5 b^3 x^6+\frac {70}{9} a^4 b^4 x^9+\frac {14}{3} a^3 b^5 x^{12}+\frac {28}{15} a^2 b^6 x^{15}+\frac {4}{9} a b^7 x^{18}+\frac {b^8 x^{21}}{21}+8 a^7 b \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^4,x]

[Out]

-1/3*a^8/x^3 + (28*a^6*b^2*x^3)/3 + (28*a^5*b^3*x^6)/3 + (70*a^4*b^4*x^9)/9 + (14*a^3*b^5*x^12)/3 + (28*a^2*b^
6*x^15)/15 + (4*a*b^7*x^18)/9 + (b^8*x^21)/21 + 8*a^7*b*Log[x]

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Maple [A]
time = 0.13, size = 90, normalized size = 0.86

method	result	size
default	\(-\frac {a^{8}}{3 x^{3}}+\frac {28 a^{6} b^{2} x^{3}}{3}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {70 a^{4} b^{4} x^{9}}{9}+\frac {14 a^{3} b^{5} x^{12}}{3}+\frac {28 a^{2} b^{6} x^{15}}{15}+\frac {4 a \,b^{7} x^{18}}{9}+\frac {b^{8} x^{21}}{21}+8 a^{7} b \ln \left (x \right )\)	\(90\)
risch	\(-\frac {a^{8}}{3 x^{3}}+\frac {28 a^{6} b^{2} x^{3}}{3}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {70 a^{4} b^{4} x^{9}}{9}+\frac {14 a^{3} b^{5} x^{12}}{3}+\frac {28 a^{2} b^{6} x^{15}}{15}+\frac {4 a \,b^{7} x^{18}}{9}+\frac {b^{8} x^{21}}{21}+8 a^{7} b \ln \left (x \right )\)	\(90\)
norman	\(\frac {-\frac {1}{3} a^{8}+\frac {1}{21} b^{8} x^{24}+\frac {4}{9} a \,b^{7} x^{21}+\frac {28}{15} a^{2} b^{6} x^{18}+\frac {14}{3} a^{3} b^{5} x^{15}+\frac {70}{9} a^{4} b^{4} x^{12}+\frac {28}{3} a^{5} b^{3} x^{9}+\frac {28}{3} a^{6} b^{2} x^{6}}{x^{3}}+8 a^{7} b \ln \left (x \right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^8/x^3+28/3*a^6*b^2*x^3+28/3*a^5*b^3*x^6+70/9*a^4*b^4*x^9+14/3*a^3*b^5*x^12+28/15*a^2*b^6*x^15+4/9*a*b^7
*x^18+1/21*b^8*x^21+8*a^7*b*ln(x)

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Maxima [A]
time = 0.30, size = 91, normalized size = 0.87 \begin {gather*} \frac {1}{21} \, b^{8} x^{21} + \frac {4}{9} \, a b^{7} x^{18} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {70}{9} \, a^{4} b^{4} x^{9} + \frac {28}{3} \, a^{5} b^{3} x^{6} + \frac {28}{3} \, a^{6} b^{2} x^{3} + \frac {8}{3} \, a^{7} b \log \left (x^{3}\right ) - \frac {a^{8}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^4,x, algorithm="maxima")

[Out]

1/21*b^8*x^21 + 4/9*a*b^7*x^18 + 28/15*a^2*b^6*x^15 + 14/3*a^3*b^5*x^12 + 70/9*a^4*b^4*x^9 + 28/3*a^5*b^3*x^6
+ 28/3*a^6*b^2*x^3 + 8/3*a^7*b*log(x^3) - 1/3*a^8/x^3

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Fricas [A]
time = 0.35, size = 94, normalized size = 0.90 \begin {gather*} \frac {15 \, b^{8} x^{24} + 140 \, a b^{7} x^{21} + 588 \, a^{2} b^{6} x^{18} + 1470 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 2940 \, a^{5} b^{3} x^{9} + 2940 \, a^{6} b^{2} x^{6} + 2520 \, a^{7} b x^{3} \log \left (x\right ) - 105 \, a^{8}}{315 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^4,x, algorithm="fricas")

[Out]

1/315*(15*b^8*x^24 + 140*a*b^7*x^21 + 588*a^2*b^6*x^18 + 1470*a^3*b^5*x^15 + 2450*a^4*b^4*x^12 + 2940*a^5*b^3*
x^9 + 2940*a^6*b^2*x^6 + 2520*a^7*b*x^3*log(x) - 105*a^8)/x^3

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Sympy [A]
time = 0.08, size = 105, normalized size = 1.00 \begin {gather*} - \frac {a^{8}}{3 x^{3}} + 8 a^{7} b \log {\left (x \right )} + \frac {28 a^{6} b^{2} x^{3}}{3} + \frac {28 a^{5} b^{3} x^{6}}{3} + \frac {70 a^{4} b^{4} x^{9}}{9} + \frac {14 a^{3} b^{5} x^{12}}{3} + \frac {28 a^{2} b^{6} x^{15}}{15} + \frac {4 a b^{7} x^{18}}{9} + \frac {b^{8} x^{21}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**4,x)

[Out]

-a**8/(3*x**3) + 8*a**7*b*log(x) + 28*a**6*b**2*x**3/3 + 28*a**5*b**3*x**6/3 + 70*a**4*b**4*x**9/9 + 14*a**3*b
**5*x**12/3 + 28*a**2*b**6*x**15/15 + 4*a*b**7*x**18/9 + b**8*x**21/21

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Giac [A]
time = 1.83, size = 100, normalized size = 0.95 \begin {gather*} \frac {1}{21} \, b^{8} x^{21} + \frac {4}{9} \, a b^{7} x^{18} + \frac {28}{15} \, a^{2} b^{6} x^{15} + \frac {14}{3} \, a^{3} b^{5} x^{12} + \frac {70}{9} \, a^{4} b^{4} x^{9} + \frac {28}{3} \, a^{5} b^{3} x^{6} + \frac {28}{3} \, a^{6} b^{2} x^{3} + 8 \, a^{7} b \log \left ({\left | x \right |}\right ) - \frac {8 \, a^{7} b x^{3} + a^{8}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^4,x, algorithm="giac")

[Out]

1/21*b^8*x^21 + 4/9*a*b^7*x^18 + 28/15*a^2*b^6*x^15 + 14/3*a^3*b^5*x^12 + 70/9*a^4*b^4*x^9 + 28/3*a^5*b^3*x^6
+ 28/3*a^6*b^2*x^3 + 8*a^7*b*log(abs(x)) - 1/3*(8*a^7*b*x^3 + a^8)/x^3

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Mupad [B]
time = 0.06, size = 89, normalized size = 0.85 \begin {gather*} \frac {b^8\,x^{21}}{21}-\frac {a^8}{3\,x^3}+\frac {4\,a\,b^7\,x^{18}}{9}+8\,a^7\,b\,\ln \left (x\right )+\frac {28\,a^6\,b^2\,x^3}{3}+\frac {28\,a^5\,b^3\,x^6}{3}+\frac {70\,a^4\,b^4\,x^9}{9}+\frac {14\,a^3\,b^5\,x^{12}}{3}+\frac {28\,a^2\,b^6\,x^{15}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^4,x)

[Out]

(b^8*x^21)/21 - a^8/(3*x^3) + (4*a*b^7*x^18)/9 + 8*a^7*b*log(x) + (28*a^6*b^2*x^3)/3 + (28*a^5*b^3*x^6)/3 + (7
0*a^4*b^4*x^9)/9 + (14*a^3*b^5*x^12)/3 + (28*a^2*b^6*x^15)/15

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